Methods and apparatus for optimal voltage and frequency control of thermally limited systems

ABSTRACT

Methods, apparatus, and articles of manufacture control a device or system that has an operational limit related to the rate or frequency of operation. The frequency of operation is controlled at a variable rate calculated to maximize the system or apparatus performance over a calculated period of time short enough that a controlling factor, such as power consumption, does not vary significantly during the period. Known system parameters, such as thermal resistance and capacitance of an integrated circuit (IC) and its package, and measured values, such as current junction temperature in an IC, are used to calculate a time-dependent frequency of operation for the upcoming time period that results in the best overall performance without exceeding the operational limit, such as the junction temperature.

This application is a continuation of U.S. patent application Ser. No.10/934,295, filed on Sep. 3, 2004, now issued as U.S. Pat. No.7,141,953, which claims the benefit of priority to U.S. ProvisionalPatent Application Ser. No. 60/500,561, filed on Sep. 4, 2003, both ofwhich are incorporated herein by reference.

TECHNICAL FIELD

Various embodiments described herein relate generally to controlling theperformance of systems having a thermal limitation that relates to theperformance rate, and more specifically to electronic devices such asintegrated circuits and other thermally limited systems such as electricmotors and combustion engines.

BACKGROUND INFORMATION

Many electronic and mechanical devices have performance limitations thatrelate to a maximum allowable temperature of operation. It is known inintegrated circuit devices (“ICs”) that higher system cycle rates resultin increased system performance, but they also result in increasedheating of the IC device. This heating may be an issue in someapplications, because it results in decreased reliability and decreasedIC lifetime. It is known to attach high thermally conductive materialsto ICs to form improved heat-dissipation structures, generally known asheat sinks, in order to increase the performance rate of the IC withoutexceeding the thermal limitation. In the case of an IC, the junctiontemperature may be the thermal limit. This thermal issue may be veryserious in certain applications, since operating an IC at a rate thatcauses the junction temperature to exceed the allowed limit for theparticular technology, results in a greatly decreased IC lifetime. Theremay be a typically exponential decrease in lifetime as a function ofsmall linear increases in junction temperature for many types of ICs, aswell as for many other systems such as electric motors.

The use of heat-dissipating devices improves the thermal limitationcapability in ICs and in other electronic devices, by reducing thetemperature difference between the outside ambient temperature and thejunction area deep within the IC. This may be known as the junction toambient temperature difference θ_(JA). Even though the performance rateof an IC can be increased without exceeding the junction temperaturethermal limit by means of a heat-dissipating device, there may still bea need to increase the performance rate to as high a level as possible.In addition, the use of heat-dissipation structures is expensive, addsyet another component subject to failure to the overall system, addsanother step to the assembly process, may require mechanical devicessuch as fans be added to the system, and may take up more space than maybe allowed in personal electronic devices.

Thus there is a need to find methods and apparatus to control theperformance rate of an electronic device to a performance level that isas high as it can be, but without exceeding the thermal limitation. Thisneed exists in electronic devices such as ICs that cannot practicallyemploy heat dissipation structures, and in ICs that use heat-dissipationstructures but need to optimize their performance rate to the bestpossible rate in order to obtain a competitive edge in the market. Theneed to control and optimize the performance rate exists in electricalsystems as well as in electronic devices, for example a powertransformer. The need to control and optimize the performance rate mayalso exist in mechanical systems as well as electrical systems, such asa motor operating an electrical generator. Any system that has a thermallimitation that relates to a controllable performance value may need tooptimize its performance under various demand levels, while notexceeding the thermal limitation at any time.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of an exemplary embodiment;

FIG. 2 is a schematic diagram of a mathematical model of an exemplaryembodiment;

FIG. 3 is a graph of time versus frequency and temperature of anexemplary embodiment;

FIG. 4 is a graph of time versus frequency and temperature of anotherexemplary embodiment;

FIG. 5 is a block diagram of an article of manufacture according tovarious embodiments of the invention, and

FIG. 6 is a flow diagram illustrating several methods according tovarious embodiments of the invention.

DETAILED DESCRIPTION

In the following detailed description of embodiments of the inventivesubject matter, reference is made to the accompanying figures that forma part thereof, and in which is shown by way of illustration, specificpreferred embodiments in which the inventive subject matter may bepracticed. These embodiments are described in sufficient detail toenable those skilled in the art to practice the inventive subjectmatter, and it is to be understood that other embodiments may beutilized and that mechanical, compositional, structural, electrical, andprocedural changes may be made without departing from the spirit andscope of the inventive subject matter. The following detaileddescription is, therefore, not to be taken in a limiting sense, and thescope of the inventive subject matter is defined only by the appendedclaims. In the drawings, like numerals describe substantially similarcomponents.

When the term “voltage” is used herein, it refers to a comparative valuein a power supply level, and the use of the term “ground” herein refersto the reference voltage level. The term “frequency”, as used herein,refers to a rate of a repeating action, such as the number of times persecond that an IC cycles through a series of operations, or the clockrate of an IC, or a number of revolutions made by a mechanical objectsuch as a rotating drive shaft.

FIG. 1 is a block diagram of an exemplary embodiment. In FIG. 1, asystem 100 having a thermal limit that depends upon a controllableperformance rate of the system is shown. Such a system mayillustratively be an IC such as a microprocessor. Microprocessors maytypically have varying workloads or amount of time that they areactively engaged in making calculations at a very high performance rate.At certain time periods the microprocessor may be operating at a maximumpossible operating rate that depends upon the technology level of themicroprocessor, the operating voltage, the clock rate, etc.

In many situations the maximum possible operating rate cannot bemaintained for a long period without the microprocessor exceeding themaximum junction temperature limit. Further, the environment in whichthe microprocessor is operating will affect how long the microprocessormay be able to operate at the maximum possible operating rate before thethermal limit is reached. For example, the microprocessor may be part ofa parallel processor system in an air-conditioned computer room, withfans blowing cooled air over the heat sink, in which case the ambienttemperature will be lower and consequently the temperature of thejunctions in the microprocessor will be lower than it would if it wereoperating in a hot environment. Thus the length of time that themicroprocessor may be able to operate at the maximum possible rate maybe longer. The same model microprocessor may alternatively be part of alaptop battery-operated computer having no fan, and operating in a hightemperature location. In this case, the length of time that themicroprocessor can operate before reaching the thermal limit will beshorter, and the thermal limit may be reached before the period of highoperating rate ends.

As a result of the above noted variations in microprocessor operatingrates depending upon the workload demands, it may be possible to allowthe microprocessor to operate at a greater performance rate than therate that would enable it to always remain below the junctiontemperature limit for short periods of time. This may be possible, forexample, if the operating rate for the preceding period of time was lowenough that the current junction temperature is well below the thermallimit at the start of the high operating rate period. This gap, betweenthe maximum allowable junction temperature and a present value ofjunction temperature, may allow the microprocessor to operate at amaximum performance rate, or at least a higher performance rate, forshort periods of time, until the time when the junction temperaturebegins to approach the limit.

A method of controlling the operating rate of a microprocessor that usesa current measurement of the microprocessor junction temperature todetermine how fast to allow the microprocessor to operate for asubsequent time period may be called dynamic thermal management (“DTM”).Several methods may be used to dynamically control the performance rateof a microprocessor, such as clock gating or power supply voltage levelreduction (known as dynamic voltage scaling, “DVS”). One method of DTMis to operate the microprocessor at the maximum possible operating rateuntil the measured junction temperature reaches the maximum allowablelevel. Then the microprocessor may be reduced in performance to anoperating level that allows the junction temperature to be lowered to asafe level, and then the microprocessor may be operated at the maximumallowable level again. In many cases this DTM process does not result inthe maximal performance over the time period that a typical high levelworkload exists for the microprocessor. The optimum rate at which tooperate a microprocessor, or any thermally limited system, will dependupon one or more external factors such as the ambient temperature, andupon one or more inherent features of the microprocessor technology,such as the thermal resistance to heat flow from the IC junction to theambient air.

FIG. 1 shows that inputting known physical parameters 102 of theparticular system (such as the thermal resistance R and the thermalcapacitance C of the IC, the maximum temperature the particular ICtechnology can sustain without undue decrease in lifetime T_(MAX), andthe maximum performance rate f_(MAX) of the technology used in the IC),in conjunction with measuring and providing certain external inputs 104(such as the temperature T, the power consumption P and the estimatedtime t_(f) that the current performance load rate will continue), to acontroller 106, it is possible to calculate an optimal time-varyingperformance rate that may optimize the performance of the system 100.This exemplary system 100 uses a simple thermal model to calculate apresent desired value of performance frequency f that will output thebest possible overall system performance for the time period t_(f),under the assumption that during the time period t_(f) the inputdemanded of the system remains relatively constant.

As used herein, “controller” means any type of computational circuit,such as but not limited to a microprocessor, a microcontroller, acomplex instruction set computing (CISC) microprocessor, a reducedinstruction set computing (RISC) microprocessor, a very long instructionword (VLIW) microprocessor, a graphics processor, a digital signalprocessor, or any other type of processor or processing circuit. In someembodiments, the functions of a controller may be performed by thedevice, apparatus, or system whose operational frequency is beingcontrolled. In other embodiments, the functions of the controller may beperformed by an independent element.

The output 108 of the controller 106 is a desired value of acontrollable performance factor (r), such as a frequency f which may becontrolled in certain illustrative embodiments of the invention byeither small increments, or essentially continuous variations, of thepower supply voltage to the illustrative microprocessor. In anotherillustrative embodiment the output 108 might alternatively be a torqueapplied to the drive wheel so f a train, and the control variable mightbe the combustible gas flow to a turbine engine. Numerous otherillustrative examples may be imagined by one of skill in the art. Thevalue of the controllable performance factor r may be determined by thefollowing equations, which will be developed in the remainder of thisdescription. In the case where r is an operating frequency, i.e., wherer=f, the frequency f=[1/α−1)kR)(T_(m)−Te^(−t/RC))/(e^(−t/RC)−e^(−(α/(α−1))t/RC))]^(1/α), where the valueα=2β+1, and where β is a constant at a given operating voltagedetermined by V=kf^(β), and k is a proportionality constant of thesystem.

FIG. 2 is a schematic diagram of a mathematical model of an exemplaryembodiment. The model uses a simple passive component electrical circuitmodel for the thermal currents that flow in a system having controllableperformance-regulating features. The thermal circuit 200 has an input202, which may be viewed as being the power consumption P(t) of thesystem 200 as a function of time. The power consumption or dissipationcontrols the temperature T(t) of the system as a function of time, sofrom a thermal point of view the input 202 may represent the current,while the junction temperature, in an illustrative embodiment of amicroprocessor or other IC device or system, may represent the voltagedifference between the points 208 and 202. The thermal capacitance C,204, and the thermal resistance R, 206, are coupled in parallel in thisillustrative embodiment of the thermal model, and the output to ground208 may represent the ambient temperature around the IC package in thisillustrative embodiment of the thermal model.

Applying Kirchoff's and Ohm's laws to the circuit of FIG. 2 results indT/dt=(P/C)−(T/τ), where τ=RC. If the relationship between the voltage Vand the frequency f in this illustrative embodiment of the system 200results in the power P being proportional to the voltage squared timesthe frequency, i.e., P=k (f*V²), and if adjusting the system voltage Vcontrols the frequency of the system 200 such that V=kf^(β), where β≧0,then the power P=kf^(α), where k is a proportionality constant that ischaracteristic of the particular technology, and where the value α=2β+1.

Solving these equations for the thermal behavior of the system 200 for amaximum allowable temperature T_(m) results in what may be called anatural frequency of the system, which is the maximum frequency at whichthe system can operate to stay exactly at the maximum allowabletemperature. This natural frequency is the highest frequency the systemcan operate at for long periods of time, and it is a function of thebasic parameters of the specific system. These parameters include thetransistor technology, the thermal resistance and capacitance of the ICand packaging, and the environment in which the system operates, such astemperature T and the power dissipation. Thus the natural frequency ofthe system is f_(nat)=(T_(m)/kR)^(1/α). If the maximum possibleoperating frequency f_(m) of a particular system is lower than thef_(nat) calculated for the current environment, then clearly the optimumoperation is at frequency f_(m) since it is lower than the naturalfrequency for the operating environment of the current time period.Therefore, we will now assume that the maximum possible frequency thatthe system is capable of achieving is greater than the natural frequencyf_(nat) of the current time period. It should be noted that the thermalmodel of this embodiment is directly applicable to many other possibleembodiments, such as combustion engines.

With such a thermal model, the optimum rate of power consumption for anyperiod of time (i.e., t_(f), known as the available time) during whichthe output performance of the system is likely to remain relativelyconstant may be calculated. Such time periods t_(f) may be very short,as in the illustrative embodiment of a microprocessor, but a knowledgeof how long typical high workload conditions last may allow sufficientlyaccurate estimates of the time period for which the optimum operatingfrequency is to be calculated.

Such an arrangement, in which the control input to the system isessentially continuously changing to improve the overall performance ofthe system during the upcoming time interval, may result in a betteraverage performance for the system as compared to a simple on/off DTMsystem. A simple on/off DTM system might have a built-in thermocouple onan IC continuously checking the junction temperature. The microprocessormight operate at a maximum possible rate until the thermal limit isdetected by the thermocouple, and then the operating rate may be dividedin half, perhaps using the system clock rate, and thus be operating at afrequency that allows the junction temperature to drop. When thejunction temperature drops to a predetermined point, the clock ratemight be increased to maximum operating rate until the thermocoupleagain indicates that the junction temperature has reached the thermallimit, and so on. Such a system would have an operating curve that lookslike a square wave of varying period. A concern with a simple on/off DTMsystem is that the lower operating rate needs to be preselected in theabsence of a method of calculating the correct lower operating speed inlight of the current external conditions and the period of time in whichthe current workload will likely continue. Thus the present arrangementmay result in improved operation over a simple on/off DTM system.

FIG. 3 is a graph of time versus frequency and temperature of anexemplary system. FIG. 3 represents a graphical representation of afirst illustrative example of the operation of a system in the casewhere the relationship of power to the frequency, P=kf^(β), is a linearrelationship, that is P=kf. This may be known as clock-throttling. Inthis case, the voltage has no impact on the frequency of operation, andthe system is controlled by setting the clock rate. The system operatesat its maximum operating frequency f_(m), 302, while the temperature ofthe junction rises at a rate or slope depending upon the thermalconductivity of the system, and of the operating environment, shown asthe dashed line 304, until the time when the temperature reaches themaximum allowable temperature T_(m), shown as the horizontal line 306.Then the clock rate is adjusted to be equal to the calculated valuef_(nat), shown as the horizontal line 308, until the end of thecalculated time period t_(f), shown as the vertical line 310.

FIG. 4 is a graph of time versus frequency and temperature of anotherexemplary embodiment. FIG. 4 represents another graphical representationof an embodiment of a more general and unconstrained system, known asthe DVS case. This illustrative example describes a situation frequentlyfound in ICs that can operate at such high frequency that the thermallimit of junction temperature may be exceeded during high workload timeperiods. In this illustrative example, the frequency of the system iscontrolled by adjusting the voltage on an essentially continuous basis,starting from a calculated initial frequency for the time period f₀, 402while the temperature of the junction in this illustrative embodiment isseen as a function of time, 404. The temperature is seen to be risingsince the initial frequency is greater than the calculated value f_(nat)408, the frequency at which the system will always stay below themaximum allowable temperature 406. As the frequency decreases, it may beseen that the temperature increases and the slope 404, decreases as itapproaches the maximum allowable temperature T_(m), shown as thehorizontal line 406. The temperature reaches the maximum level 406 atthe same time that the frequency reaches the calculated value f_(nat),408, at a time labeled t_(th), known as the threshold time 412, whichcontinues until the end of the calculated time period 410.

It should be noted that if the device monitoring the system powerdissipation, such as the external inputs 104 from FIG. 1, measures apower level that has shifted by more than a specified amount, such as10%, from the power value at the start of the time period, then therecalculation of the frequency may be initiated immediately, rather thanwaiting for the end of the calculated time period t_(f), 410. It shouldalso be noted that the variation of the frequency may be continuous,even though FIG. 4 may show abrupt changes of slope at certain points inthe figure, and that in the general case the frequency may decline fromthe initial frequency at an exponential rate by the factore^(−t/((α−1)RC)), thus depending on the thermal resistance andcapacitance of the system. It should also be noted that an estimation ofthe rate of frequency decline may be obtained by use of a linearapproximation in cases where the voltage is linearly related to thefrequency. The rate of frequency decline may alternatively be obtainedin non-linear cases by use of the Taylor expansion series with anaccuracy of 0.04, or 4%.

FIG. 5 is a block diagram of an article of manufacture 502 according tovarious embodiments of the invention. The article of manufacture 502 maycomprises one or more of a number of possible elements, such as acommunications network, a computer, a memory system, a magnetic oroptical disk, some other information storage device, and/or any type ofelectronic device or system. The article 502 may comprise at least oneprocessor 504 coupled to a machine-accessible medium such as a memory506, storing associated information (e.g., computer program instructions508, and/or other data), and an input/output driver 510 coupled to anexternal electrical device by various elements, such as a bus or cable512, which when accessed, results in a machine performing such actionsas calculating a solution to a mathematical problem. Various ones of theelements of the article 502, for example the processor 504, may havethermal limitation issues and may use embodiments of the invention tohelp alleviate and moderate the thermal situation by controlling theoperating rate at the optimal frequency. As an illustrative example, theprocessor 504 may be arranged with an onboard temperature measuringdevice such as a diode junction, and with an onboard power dissipationmeasuring device, and it thus may be able to calculate the mostefficient operating voltage to control the frequency of the processor504, using methods such as those discussed and shown previously in FIGS.3 and 4. With such an arrangement, the processor 504 may respond to thevarying calculation requirements of the article 502 at the maximumpossible rate compatible with not exceeding the junction temperaturelimit.

Alternatively, the article 502 may comprise a portion or an element of acommunications network in two-way communications with other elements ofthe network by means of the bus or cable 512, or by wirelesscommunications elements included in I/O driver 510, or use both cableand wireless elements. In this illustrative example of an element of acommunications network, the two-way wireless communications apparatusmay include a dipole antenna, a monopole antenna, a unidirectionalantenna, a laser infrared “IR” diode emitter/detector, or any othersuitable type of communication structure. The processor 504 may acceptsignals from the I/O driver 510 and perform an operation under thecontrol of a program in memory 506, or computer program instructions508.

FIG. 6 is a flow diagram illustrating several methods according tovarious embodiments.

In 602, the allowable power consumption range, along with inherentdevice values for thermal resistance and capacitance, and a maximumallowable temperature, are determined. These values may be provided, forexample, using one or more controllers and/or information storage units.

In 604, selected present operating parameters, such as devicetemperature, are determined. Other present operating parameters, such asvoltage level, current usage, and the operating frequency, may also bedetermined. The selected operating parameters may be determined in anysuitable manner and through any suitable element(s) or unit(s). Forexample, they may be provided by suitable sensors, meters, or gauges,and/or by accessing them from one or more memory elements.

In 606, the power consumption and a time interval during which the powerconsumption will remain within a fixed range of the present value arecalculated. This may be performed, for example, using one or morecontrollers and/or information storage units.

In 608, the present value of measured temperature is compared with themaximum allowable temperature value. If the present temperature value isequal to or greater than the maximum allowable temperature value, themethod goes to 610.

In 610, a selected algorithm to handle temperature emergencies isapplied, such as clock throttling. The method flow then returns to 604and continues until such time as the present value of measuredtemperature may be determined at 608 to be below the stored maximumallowable temperature value, in which event the method flow goes to 612.

In 612, a new optimum operating frequency as a function of time for thecalculated time interval of 606 is calculated.

In 614, the voltage as a function of time required to operate the deviceat the calculated optimum frequency as a function of time is calculated.The method then returns to 604.

It should be noted that the individual activities shown in the flowdiagrams do not have to be performed in the order illustrated or in anyparticular order. Moreover, various activities described with respect tothe methods identified herein can be executed in serial or parallelfashion. Some activities may be repeated indefinitely, and others mayoccur only once. Various embodiments may have more or fewer activitiesthan those illustrated.

There are numerous other devices and systems that may benefit from theuse of the described embodiments. Any system having a performance ratecontrollable by a combination of inputs, such as input voltage andcurrent, and a limitation that is a measurable quantity may use thismethod. The inventive subject matter has been described using a simpleillustrative example of a microprocessor with a varying workload and amaximum operation rate that will cause the microprocessor to eventuallyoverheat, depending upon the system parameters and environment. Thedisclosed subject matter is not so limited, and it may be applied toother thermally limited systems and to systems having non-thermallimits. Examples of such systems include, but are not limited to,electric motors driving a train, internal combustion engines drivingvariable loads, turbine engines and steam turbines driving electricalgenerators, and a rocket engine.

The accompanying figures that form a part hereof show by way ofillustration, and not of limitation, specific embodiments in which theinventive subject matter may be practiced. The embodiments illustratedare described in sufficient detail to enable those skilled in the art topractice the teachings disclosed herein. Other embodiments may beutilized and derived therefrom, such that structural and logicalsubstitutions and changes may be made without departing from the scopeof this disclosure. This Detailed Description, therefore, is not to betaken in a limiting sense, and the scope of various embodiments isdefined only by the appended claims, along with the full range ofequivalents to which such claims are entitled.

Such embodiments of the inventive subject matter may be referred toherein, individually or collectively, by the term “invention” merely forconvenience and without intending to voluntarily limit the scope of thisapplication to any single invention or inventive concept if more thanone is in fact disclosed. Thus, although specific embodiments have beenillustrated and described herein, it should be appreciated that anyarrangement calculated to achieve the same purpose may be substitutedfor the specific embodiments shown. This disclosure is intended to coverany and all adaptations or variations of the various embodiments.Combinations of the above embodiments, and other embodiments notspecifically described herein, will be apparent to those of skill in theart upon reviewing the above description.

The Abstract of the Disclosure is provided to comply with 37 C.F.R.§1.72(b), requiring an abstract that will allow the reader to quicklyascertain the nature of the technical disclosure. It is submitted withthe understanding that it will not be used to interpret or limit thescope of meaning of the claims. In addition, in the foregoing DetailedDescription, it may be seen that various features are grouped togetherin a single embodiment for the purpose of streamlining this disclosureand increasing its clarity. This method of disclosure is not to beinterpreted as reflecting an intention that the claimed embodimentsrequire more features than are expressly recited in each claim. Rather,as the following claims reflect, inventive subject matter lies in lessthan all features of a single disclosed embodiment. Thus the followingclaims are hereby incorporated into the Detailed Description, with eachclaim standing on its own as a separate embodiment.

1. A method comprising: determining operating parameters in a systemhaving a thermal constraint that is dependent upon a controllableperformance rate of the system; further comprising optimizing theperformance rate while not exceeding the thermal constraint; wherein thesystem includes at least one of a semiconductor device, a powertransformer, a steam engine, an electric motor, a turbine, and aninternal combustion engine; wherein the controllable performance rate ofthe system includes at least one of an output voltage, an outputcurrent, an output power, revolutions per minute, an output pressure,operations per second, calculations per second and operating frequency;and wherein the performance rate is controlled by at least one of anoperating voltage level, an operating current level, an input powerlevel, a clock rate, a thermal input flow rate, an input fuel flow rateand an input oxidizer flow rate; wherein further a natural operatingfrequency (f_(nat)) of the system operating at a maximum allowabletemperature (T_(m)) is calculated for a current set of operatingconditions including a thermal resistance (R), a thermal capacitance(C), a voltage (V), an initial temperature (T₀) and a power consumption(P) of the system, wherein f_(nat)=(T_(m)/k R)^(1/α), where α is anexponent of the operating frequency (f) of the system proportional tothe power consumption (P) and a system proportionality constant (k),where P=kf^(α).
 2. An article comprising a machine-accessible mediumhaving associated instructions, wherein the instructions, when accessedby a machine, result in the machine performing the following actions:determining an operating parameter in a system having a thermalconstraint that is dependent upon a controllable performance rate of thesystem; further including an electronic system comprising amicroelectronic device, the thermal constraint comprises a maximumjunction temperature (T_(MAX)), the controllable performance ratecomprises an operating frequency (f), and wherein the instructions, whenaccessed, further result in the machine performing: controlling aperformance of the system by calculating f as a function of operatingparameters including time (t), temperature (T), maximum temperature(T_(m)), system power (P), voltage (V), thermal resistance (R), andthermal capacitance (C), and by recalculating f at predetermined timeintervals or if P changes by a predetermined amount; further comprisingthe electronic system calculating the operating frequency determined byf=[(1/(α−1)kR)(T_(m)−T e^(−t/RC))/(e^(−t/RC)−e^(−(α/(α−1))t/RC))]^(1/α),where the value α=2β+1, and where β is a constant at a given operatingvoltage determined by V=kf^(β), and k is a proportionality constant ofthe system.
 3. A method of controlling a system, comprising: determiningoperating parameters in a system having a thermal constraint that isdependent upon a controllable performance rate of the system; optimizingthe performance rate while not exceeding the thermal constraint bycalculating a present performance rate at regular time intervalsdetermined by a typical time for the system to remain at a stabletemperature; and adjusting the performance rate by controlling an inputvariable selected from a list including voltage, current, power, fuelflow rate, oxidizer flow rate, thermal energy flow rate and clock rate;wherein the thermal constraint is directly measurable and includes atleast one of a junction temperature, a combustion chamber temperature, amaximum conductor temperature, and a maximum case temperature maximumjunction temperature T_(Jmax); the controllable performance rate (r)comprises at least one of an operating frequency (f) an output voltage,an output current, an output power, revolutions per minute, an outputpressure, operations per second, and calculations per second; andcontrolling the performance of the system comprises calculating theperformance rate (r) as a function of operating parameters includingtime (t), present temperature (T), maximum temperature (T_(m)), systempower (P), voltage (V), thermal resistance (R) and thermal capacitance(C), and recalculating the performance rate if measured temperaturechanges by a predetermined amount, wherein voltage includes one ofelectrical voltage and system pressure; wherein calculating theperformance rate (r) is determined by r=[(1/(α−1)kR)(T_(m)−Te^(−t/RC))/(e^(−t/RC)−e^(−(α/(α−1))t/RC))]^(1/α), where the valueα=2β+1, and where β is a constant at a given operating voltagedetermined by V=k r^(β), where β≧0, and k is a proportionality constantof the system.